E.2 Continuous Distrbutions
| Uniform | \(U(a,b)\) |
|---|---|
| Parameters: | a = minimum, b = maximum, \(-\infty < a < b < \infty\) |
| PDF: | \(f(x) = \frac{1}{b-a}\) for \(a \leq x \leq b\) |
| CDF: | \(F(x) = \frac{x-a}{b-a} \; \text{if} \; a \leq x \leq b\) |
| Inverse CDF: | \(F^{-1}(p) = a + p(b-a) \; \; \text{if} \; 0 < p < 1\) |
| Expected Value: | \(E[X]=\frac{a+b}{2}\) |
| Variance: | \(V[X] = \frac{(b-a)^2}{12}\) |
| Generation: | UniformRV(min: Double, max:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | = a + RAND()*(b-a) |
| Modeling: | assumes equally likely across the range, |
| when you have lack of data, task times |
| Normal | \(N(\mu,\sigma^2)\) |
|---|---|
| Parameters: | \(-\infty < \mu < +\infty\) (mean), \(\sigma^2 > 0\) (variance) |
| CDF: | No closed form |
| Inverse CDF: | No closed form |
| Expected Value: | \(E[X] = \mu\) |
| Variance: | \(Var[X] = \sigma^2\) |
| Generation: | NormalRV(mean:Double, variance:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | = NORM.INV(RAND(), \(\mu\), \(\sigma\)) |
| Modeling: | task times, errors |
| Exponential | EXPO(\(1/\lambda\)) |
|---|---|
| Parameters: | \(\lambda > 0\) |
| PDF: | \(f(x) = \lambda e^{-\lambda x} \; \text{if} \; x \geq 0\) |
| CDF: | \(F(x) = 1 - e^{-\lambda x} \; \text{if} \; x \geq 0\) |
| Inverse CDF: | \(F^{-1}(p) = (-1/\lambda)\ln \left(1-p \right) \; \; \text{if} \; 0 < p < 1\) |
| Expected Value: | \(E[X] = \theta = 1/\lambda\) |
| Variance: | \(Var[X] = 1/\lambda^2\) |
| Generation: | ExponentialRV(mean:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | = \((-1/\lambda)\)LN(1-RAND()) |
| Modeling: | time between arrivals, time to failure |
| highly variable task time |
| Weibull | WEIB(\(\beta\), \(\alpha\)) |
|---|---|
| Parameters: | \(\beta > 0\) (scale), \(\alpha > 0\) (shape) |
| CDF: | \(F(x) = 1- e^{-(x/\beta)^\alpha} \; \text{if} \; x \geq 0\) |
| Inverse CDF: | \(F^{-1}(p) = \beta\left[ -\ln (1-p)\right]^{1/\alpha} \; \; \text{if} \; 0 < p < 1\) |
| Expected Value: | \(E[X] = \left(\dfrac{\beta}{\alpha}\right)\Gamma\left(\dfrac{1}{\alpha}\right)\) |
| Variance: | \(Var[X] = \left(\dfrac{\beta^2}{\alpha}\right)\biggl\lbrace 2\Gamma\left(\dfrac{2}{\alpha}\right) - \left(\dfrac{1}{\alpha}\right)\biggl(\Gamma\left(\dfrac{1}{\alpha}\right)\biggr)^2\biggr\rbrace\) |
| Generation: | WeibullRV(shape:Double, scale:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | = \((\beta)(-\text{LN}(1-\text{RAND}())\wedge(1/\alpha)\) |
| Modeling: | task times, time to failure |
| Erlang | Erlang(\(r\),\(\beta\)) |
|---|---|
| Parameters: | \(r > 0\), integer, \(\beta > 0\) (scale) |
| CDF: | \(F(x) = 1- e^{(-x/\beta)}\sum\limits_{j=0}^{r-1}\dfrac{(x/\beta)^j}{j} \; \text{if} \; x \geq 0\) |
| Inverse CDF: | No closed form |
| Expected Value: | \(E[X] = r\beta\) |
| Variance: | \(Var[X] = r\beta^2\) |
| Generation: | use GammaRV class |
| Spreadsheet Generation: | = GAMMA.INV(RAND(), \(r\), \(\beta\)) |
| Modeling: | task times, lead time, time to failure, |
| Gamma | Gamma(\(\alpha\),\(\beta\)) |
|---|---|
| Parameters: | \(\alpha > 0\), shape, \(\beta > 0\) (scale) |
| CDF: | No closed form |
| Inverse CDF: | No closed form |
| Expected Value: | \(E[X] = \alpha \beta\) |
| Variance: | \(Var[X] = \alpha \beta^2\) |
| Generation: | GammaRV(shape:Double, scale:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | = GAMMA.INV(RAND(), \(\alpha\), \(\beta\)) |
| Modeling: | task times, lead time, time to failure, |
| Beta | BETA(\(\alpha_1\),\(\alpha_2\)) |
|---|---|
| Parameters: | shape parameters \(\alpha_1 >0\), \(\alpha_2 >0\) |
| CDF: | No closed form |
| Inverse CDF: | No closed form |
| Expected Value: | \(E[X] = \dfrac{\alpha_1}{\alpha_1 + \alpha_2}\) |
| Variance: | \(Var[X] = \dfrac{\alpha_1\alpha_2}{(\alpha_1 + \alpha_2)^2(\alpha_1 + \alpha_2+1)}\) |
| Generation: | BetaRV(alpha1:Double, alpha2:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | BETA.INV(RAND(), \(\alpha_1\), \(\alpha_2\)) |
| Modeling: | activity time when data is limited, probabilities |
| Lognormal | LOGN\(\left(\mu_l,\sigma_l\right)\) |
|---|---|
| Parameters: | \(\mu = \ln\left(\mu_{l}^{2}/\sqrt{\sigma_{l}^{2} + \mu_{l}^{2}}\right) \quad \sigma^{2} = \ln\left((\sigma_{l}^{2}/\mu_{l}^{2}) + 1\right)\) |
| CDF: | No closed form |
| Inverse CDF: | No closed form |
| Expected Value: | \(E[X] = \mu_l = e^{\mu + \sigma^{2}/2}\) |
| Variance: | \(Var[X] = \sigma_{l}^{2} = e^{2\mu + \sigma^{2}}\left(e^{\sigma^{2}} - 1\right)\) |
| Generation: | LognormalRV(mean:Double, variance:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | LOGNORM.INV(RAND(), \(\mu\), \(\sigma\)) |
| Modeling: | task times, time to failure |
| Triangular | TRIA(a, m, b) |
|---|---|
| Parameters: | a = minimum, m = mode, b = maximum |
| CDF: | \(F(x) = \dfrac{(x - a)^2}{(b - a)(m - a)} \; \text{for} \; a \leq x \leq m\) |
| \(F(x) = 1 - \dfrac{(b - x)^2}{(b - a)(b - m)} \; \text{for} \;m < x \leq b\) | |
| Inverse CDF: | \(F^{-1}(u) = a + \sqrt{(b-a)(m-a)u} \; \text{for} \; 0 < u < \dfrac{m-a}{b-a}\) |
| \(F^{-1}(u) = b - \sqrt{(b-a)(b-m)(1-u)} \; \text{for} \; \dfrac{m-a}{b-a} \leq u\) | |
| Expected Value: | \(E[X] = (a+m+b)/3\) |
| Variance: | \(Var[X] = \dfrac{a^2 + b^2 + m^2 -ab -am -bm}{18}\) |
| Generation: | TriangularRV(min:Double, mode:Double, max:Double, stream:RNStreamIfc) |
| Spreadsheet Generation: | implement \(F^{-1}(u)\) as VBA function |
| Modeling: | task times, activity time when data is limited |